Question

For a fixed amount of gas held at constant pressure, calculate the temperature in degrees Celsius to which the gas would have to be changed to achieve the change in volume shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Temperature } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Final } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Final } \\ \text { Temperature } \end{array} \\ \hline 100.0^{\circ} \mathrm{C} & 250.0 \mathrm{~mL} & 100.0 \mathrm{~mL} & ? \\\ \hline 27.5^{\circ} \mathrm{C} & 125 \mathrm{~mL} & 148 \mathrm{~mL} & ? \\ \hline 300 \mathrm{~K} & 13.7 \mathrm{~L} & 57.2 \mathrm{~L} & ? \\ \hline \end{array} $$

Short answer

The final temperatures after the changes in volume are -123.89°C, 84.23°C, and 980.12°C respectively.

Step-by-step solution

Convert Initial Temperature to Kelvin

In the case where the temperature is given in Celsius, it needs to be first converted to Kelvin. The formula to convert Celsius to Kelvin is \( K = °C + 273.15 \). Thus for the first query in the table, the initial temperature would be \( 100.0°C + 273.15 = 373.15K \) and for the second query, it would be \( 27.5°C + 273.15 = 300.65K \). For the third query, the temperature is already given in Kelvin.

Apply Charles' Law

The next step is to apply Charles' Law, which is \( V1/T1 = V2/T2 \). Solving for the final temperature \( T2 \), the equation becomes \( T2 = V2 * T1 / V1 \). Applying that for each query of the table gives the final temperatures: for the first query \( T2 = (100.0mL * 373.15K) / 250.0mL = 149.26K \), for the second query \( T2 = (148mL * 300.65K) / 125mL = 357.38K \), and for the third query \( T2 = (57.2L * 300K) / 13.7L = 1253.27K \).

Convert Final Temperatures to Degrees Celsius (Optional)

The final temperatures calculated are in Kelvin. If the final temperatures are desired in Celsius, they need to be converted using the formula \( °C = K - 273.15 \). Thus, for the first query the final temperature would be \( 149.26K - 273.15 = -123.89°C \), for the second query it would be \( 357.38K - 273.15 = 84.23°C \), and for the third query it would be \( 1253.27K - 273.15 = 980.12°C \).

Key concepts

Understanding Gas Laws

Gas laws, such as Charles' Law, are fundamental principles in chemistry that describe the behavior of gases. Charles' Law specifically deals with the relationship between volume and temperature when pressure is held constant. This law states that the volume of a gas is directly proportional to its absolute temperature (measured in Kelvin) if the pressure remains unchanged. This means, if the temperature of a gas increases, its volume will also increase, provided the pressure doesn't change, and vice versa.

Here's how it works:

• Given a fixed amount of gas, if you increase the temperature, the gas particles gain more energy, move faster, and require more space, thus increasing the volume.
• Conversely, if you decrease the temperature, the particles move slower, take up less space, and the volume decreases.

When using gas laws, always remember to work with absolute temperatures (Kelvin). This ensures the proportionality is maintained, as Kelvin is an absolute scale starting from zero, unlike Celsius or Fahrenheit.

The Importance of Temperature Conversion

Temperature conversion is a critical step when working with gas laws because these laws always require temperatures to be in Kelvin. The Kelvin scale is preferred over Celsius or Fahrenheit because it starts at absolute zero, the point at which molecular motion ceases.

To convert Celsius to Kelvin:

• Use the formula: \( K = °C + 273.15 \).
• This calculation ensures that you have the correct units for temperature in your equations.

When you encounter a temperature in a gas law problem, always check its unit. If it's in Celsius, convert it to Kelvin before performing any calculations. This step is crucial because using temperatures in Celsius or Fahrenheit could lead to incorrect results, as these scales do not start at absolute zero. Always double-check your conversion to avoid errors.

Exploring the Volume and Temperature Relationship

The relationship between volume and temperature in gases is elegantly expressed by Charles' Law, which can be mathematically represented as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). This equation tells us that the ratio of the initial volume and temperature (in Kelvin) is equal to the ratio of the final volume and temperature.

When applying this law:

• Identify the initial and final volumes and the initial temperature (converted to Kelvin) from the problem.
• Use these values to solve for the unknown final temperature or final volume.

To isolate the final temperature \( T_2 \), rearrange the equation to \( T_2 = \frac{V_2 \times T_1}{V_1} \). This allows you to calculate the necessary change in temperature needed to achieve a desired change in volume, under constant pressure. Remember, this relationship holds true only when pressure remains constant, highlighting the necessity of understanding conditions when applying gas laws to real-world scenarios.